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        <title>Abelian Group</title>

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                <h1 id="title" titleSize="">
                    Abelian Group
                </h1>
            
            <h1 id="motivation--definition">Motivation &amp; Definition</h1>
<p>Here, we consider a class of groups which $-$ despite being very general $-$ exhibit very useful properties absent in general groups.</p>
<div class="env envDef" id=""><img class="icon noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/css/fa/definition.svg"><b class="envTitle">Definition. </b>A group $G$ is said to be <em>Abelian</em> if $gh=hg$ for all $g,h\in G$.</div>

<h2 id="abelian-groups-as-hahahugoshortcode8s1hbhb">Abelian Groups as <a href=https://zhaoshenzhai.github.io/mathwiki/module.md class="internalLink references" title="Module" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/module.md&#34;, {&#34;Date&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Module&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/module&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Module&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/module&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/module.md&#34;, {&#34;Date&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-06T18:07:45-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Module&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/module&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">$\Z$-modules</a></h2>
<p>We denote the <a href=https://zhaoshenzhai.github.io/mathwiki/subcategory.md class="internalLink references ghostLink" title="subcategory" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/subcategory.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/subcategory.md&#34;, &#34;nopPage&#34;);">subcategory</a> of Abelian groups as $\catabgrp$, which is one of the most well-behaved categories. Indeed, <span style="color:red">???</span>.</p>
<ul>
<li>We have a forgetful functor $\catmod[\Z]\to\catabgrp$, which is an isomorphism of categories <span style="color:red">???</span>. In this way, the notions of <a href=https://zhaoshenzhai.github.io/mathwiki/torsion.md class="internalLink constructions ghostLink" title="torsion" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/torsion.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/torsion.md&#34;, &#34;nopPage&#34;);">torsion</a>, <a href=https://zhaoshenzhai.github.io/mathwiki/rank.md class="internalLink constructions ghostLink" title="rank" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/rank.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/rank.md&#34;, &#34;nopPage&#34;);">rank</a>, etc, all carry forward.</li>
</ul>
<h2 id="the-category-catabgrp">The Category $\catabgrp$</h2>
<h1 id="general-techniques">General Techniques</h1>
<h2 id="structure-theory">Structure Theory</h2>
<p>Since Abelian groups are just $\Z$-modules and $\Z$ is a <a href=https://zhaoshenzhai.github.io/mathwiki/principal_ideal_domain.md class="internalLink references" title="Principal Ideal Domain" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/principal_ideal_domain.md&#34;, {&#34;Date&#34;:&#34;2024-08-31T19:14:26-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-31T19:14:26-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-31T19:14:26-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Principal Ideal Domain&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/principal_ideal_domain&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-08-31T19:14:26-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-31T19:14:26-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-31T19:14:26-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Principal Ideal Domain&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/principal_ideal_domain&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/principal_ideal_domain.md&#34;, {&#34;Date&#34;:&#34;2024-08-31T19:14:26-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-31T19:14:26-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-31T19:14:26-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Principal Ideal Domain&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/principal_ideal_domain&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">PID</a>, the <a href=https://zhaoshenzhai.github.io/mathwiki/structure_theorem_for_fg_modules_over_pid.md class="internalLink properties" title="Structure Theorem for Finitely-generated Modules over PIDs" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/structure_theorem_for_fg_modules_over_pid.md&#34;, {&#34;Date&#34;:&#34;2024-09-04T12:12:19-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-09-04T12:12:19-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-09-04T12:12:19-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Structure Theorem for Finitely-generated Modules over PIDs&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/structure_theorem_for_fg_modules_over_pid&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-09-04T12:12:19-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-09-04T12:12:19-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-09-04T12:12:19-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Structure Theorem for Finitely-generated Modules over PIDs&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/structure_theorem_for_fg_modules_over_pid&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/structure_theorem_for_fg_modules_over_pid.md&#34;, {&#34;Date&#34;:&#34;2024-09-04T12:12:19-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-09-04T12:12:19-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-09-04T12:12:19-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Structure Theorem for Finitely-generated Modules over PIDs&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/structure_theorem_for_fg_modules_over_pid&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">Structure Theorem</a> for finitely-generated modules over a PID applies:</p>
<div class="env" id=""><img class="icon noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/css/fa/theorem.svg"><b class="envTitle">Theorem (Structure Theorem for f.g. Abelian Groups). </b>Every f.g. Abelian group $G$ admits a decomposition $G\iso\Z^{\rk G}\oplus\Tor G$, and furthermore
$$\begin{equation}
\Tor G\iso\bigoplus_{i,j}\Z/\mf{p}_i^{\alpha_{ij}}\iso\bigoplus_j\Z/\mf{a}_j
\end{equation}$$
for unique (up to a permutation) prime ideals $\mf{p}_i\nsubgrp\Z$ <span style="color:gray">(elementary divisors)</span> and positive integers $\alpha_{ij}\in\N$, where $\mf{a}_j\coloneqq\prod_i\mf{p}_i^{\alpha_{ij}}$ <span style="color:gray">(invariant factors)</span>.</div>

<p>On the other hand, Abelian groups that are <em>not</em> finitely-generated are much wilder, and many set-theoretical issues crop up.</p>
<h1 id="examples">Examples</h1>
<h2 id="hahahugoshortcode8s8hbhb"><a href=https://zhaoshenzhai.github.io/mathwiki/cyclic_group.md class="internalLink examples" title="Cyclic Group" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/cyclic_group.md&#34;, {&#34;Date&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Cyclic Group&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/cyclic_group&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Cyclic Group&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/cyclic_group&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/cyclic_group.md&#34;, {&#34;Date&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-03T14:23:46-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Cyclic Group&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/cyclic_group&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">Cyclic Groups</a></h2>
<h1 id="generalizations">Generalizations</h1>
<h2 id="abelian-categories">Abelian Categories</h2>


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                May 22, 2024 | Zhaoshen Zhai

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                        <div class="metaButton metaRef listenDark">#Alu09</div>
                    
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